Problem 39
Question
Solve each exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$7^{0.3 x}=813$$
Step-by-Step Solution
Verified Answer
The solution for \(x\) in terms of natural logarithms is \(x = \frac{\ln(813)}{0.3 \ln(7)}\). As for the decimal approximation of \(x\), it must be calculated using a calculator.
1Step 1: Apply the natural logarithm
Use logarithmic form to write the given equation. It becomes: \(\ln(7^{0.3x}) = \ln(813)\).
2Step 2: Use the properties of logarithms
Apply the power rule of logarithms to bring the exponent out front: \(0.3x \times \ln(7) = \ln(813)\).
3Step 3: Solve for x
Isolate \(x\) by dividing both sides of the equation by \(0.3 \ln(7)\). This results in \(x = \frac{\ln(813)}{0.3\ln(7)}\).
4Step 4: Calculate decimal approximation
Use a calculator to determine the decimal approximation of \(x\).
Other exercises in this chapter
Problem 38
Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.
View solution Problem 39
Evaluate each expression without using a calculator. $$\log _{5} 5^{7}$$
View solution Problem 39
Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.
View solution Problem 40
Evaluate each expression without using a calculator. $$\log _{4} 4^{6}$$
View solution